12 THE SPACE L°° 173 Suppose that M^/) < +00. Then for each a> M^/) the set of points x e X such that f(x) > a is negligible. Now the set of points jc e X such that f(x) > M00(/) is the union of the sets of jc e X such that f(x) > rn, where (rj is a decreasing sequence with M^/) as limit. Hence (13.6.2) we have m^f) ^f(x) ^ M^/) almost everywhere in X. It follows that mn(f)£UJJ) if /x # 0. (If IJL = 0, then mj/) = +00 and MJ/) = -oo.) The relation m^f) = M^/) is equivalent to saying that /is equivalent to a constant if ^ ^ 0. Also (still assuming that ju 7^ 0) we have inf/(x) g ess inf/(x) ^ ess sup/(x) ^ sup/(x). xeX jceX xeX JceX If two functions /, g are equivalent, then "«(/) = ^oo (^) and M, (/) = M^). Hence we can define mm(f) and M^/) for a function /which is defined only almost everywhere in X: we choose any function g e/such that g is defined everywhere in X, and put mm(f) = mj(g) and Mn(f) = M^g). If /and g are such that/+ g is defined almost everywhere, then (13.12.1) MJ/+ g) g MJ/) -f MUtf) wherever the right-hand side is defined. Likewise, if /and ^ are both ^ 0, then with the product convention of (13.11). A function /, defined almost everywhere on X, with values in R or C, is said to be bounded in measure or essentially bounded (with respect to /i) if M^d/l) < -f oo ; it follows that/is then finite almost everywhere. A bounded function is bounded in measure. (13.12.2) (Mean value theorem) Let f '• X->R be measurable and bounded in measure. For every integrable function g ^ 0, the function fg (which is defined and finite almost everywhere) is integrable, and (13.12.2.1) mj/) (g d» £ (fg d^ ^ MM) \9 dp* Furthermore •, two of the three terms in (13.12.2.1) are equal only if, in the (measurable) set S of points xe X such that g(x) ^ 0, we have either f(x) = MO^/) almost everywhere, or f(x) = m^f) almost everywhere.) = -MJ-/).h satisfy the